The idea of the ABB stepsize is to take a larger step when \(\cos ^2(\mathbf {s}_{k-1}, \mathbf {y}_{k-1}) \approx 1\) , and a smaller step when this is not the case. If \(\cos ^2(\mathbf {s}_{k-1}, \mathbf {y}_{k-1})\)  is close to 1, this means that \(\mathbf {g}_{k-1}\)  (or, equivalently, \(\mathbf {s}_{k-1}\) ) is close to an eigenvector of \(A\)  corresponding to an eigenvalue \(\lambda > 0\) . Thus, as we will see in Section , by the step \(\beta _{k}^{\rm {BB1}}\)  we are particularly reducing the gradient component corresponding to \(\lambda \)  significantly. When we are far from an eigenvalue, we prefer to take shorter steps, such as \(\beta _{k}^{\rm {BB2}}\) , since we aim to reduce several gradient components; this fosters the gradient method to take a new longer step in the next iterations.
There exist several variants of the ABB method; the interested reader may refer to [1]}, [2]} for such ideas.
